A218482 First differences of the binomial transform of the partition numbers (A000041).

1, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812, 309274516740

Offset

0,3

Comments

a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.

From Gus Wiseman, Dec 12 2022: (Start)

Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:

() ((1)) ((2)) ((3))

((11)) ((12))

((1)(1)) ((21))

((111))

((1)(2))

((2)(1))

((11)(1))

((1)(1)(1))

The case of constant lengths is A101509.

The case of strictly decreasing lengths is A129519.

The case of sequences of partitions is A141199.

The case of twice-partitions is A358831.

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).

G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).

G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.

G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).

G.f.: exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).

G.f.: Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.

a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

Example

G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
The g.f. equals the product:
A(x) = (1-x)/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...

Maple

b:= proc(n) option remember;
      add(combinat[numbpart](k)*binomial(n, k), k=0..n)
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2014

Mathematica

Flatten[{1, Table[Sum[Binomial[n-1, k]*PartitionsP[k+1], {k, 0, n-1}], {n, 1, 30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

Prog

(PARI) {a(n)=sum(k=0, n, (binomial(n, k)-if(n>0, binomial(n-1, k)))*numbpart(k))}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(prod(k=1, n, (1-x)^k/((1-x)^k-X^k)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(sum(m=0, n, x^m*(1-x)^(m*(m-1)/2)/prod(k=1, m, ((1-x)^k - X^k))), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(sum(m=0, n, x^(m^2)*(1-X)^m/prod(k=1, m, ((1-x)^k - x^k)^2)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(exp(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(exp(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(prod(k=1, n, (1 + x^k/(1-X)^k)^valuation(2*k, 2)), n)}

Crossrefs

Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.

Sequence in context: A030015 A318567 A103446 * A094723 A127358 A077849

Adjacent sequences: A218479 A218480 A218481 * A218483 A218484 A218485

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2012

Status

approved